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- Homework Statement
- Find the area of the unit sphere in R3 enclosed in the offset cone (x-1)^2 +y^2 = z^2

- Relevant Equations
- x^2 + y^2 + z^2 = 1

(x-1)^2 +y^2 = z^2

Problem: The sphere is parametrized in cylindrical coordinates by:

x = r cosθ

y = r sinθ

z = (1-r^2)^1/2

and intersected by the cone (x-1)^2 +y^2 = z^2.

find the area of the sphere enclosed by the cone using the equation:

da = r/(1-r^2) dr dθ

Attempt at solution:

from the equations for the sphere and cone: r = cosθ describes the intersection in the relevant coordinates.

the values of r ranges from 0 to 1, and θ from -pi/2 to pi/2.

How does one set the limits of integration for the area integral using da = r/(1-r^2) dr dθ ?

I tried setting r from 0 to cosθ and kept getting pi as the area, which is too large.

x = r cosθ

y = r sinθ

z = (1-r^2)^1/2

and intersected by the cone (x-1)^2 +y^2 = z^2.

find the area of the sphere enclosed by the cone using the equation:

da = r/(1-r^2) dr dθ

Attempt at solution:

from the equations for the sphere and cone: r = cosθ describes the intersection in the relevant coordinates.

the values of r ranges from 0 to 1, and θ from -pi/2 to pi/2.

How does one set the limits of integration for the area integral using da = r/(1-r^2) dr dθ ?

I tried setting r from 0 to cosθ and kept getting pi as the area, which is too large.